1. Field of the Invention
The present invention relates to a measuring device and a measuring method for measuring variations in electric characteristics of a semiconductor device.
2. Description of the Background Art
Along with size reduction of a semiconductor device, relative variations in electric characteristics among elements are becoming greater. Such variations in characteristics cause a malfunction of a circuit and a reduction in yields of products. Therefore, there is a growing need for achieving circuit design taking such variations into account upon correct estimation of the variations.
As statistical analysis means for estimating variations quantitatively, there is a method of measuring a plurality of samples under the same condition and calculating a standard deviation of measured results. In this case, a measurement on a certain electric characteristic of the plurality of samples results in a measured value that generally includes uncertainties due to an external noise and the like in addition to a characteristic amount desired to be measured.
In many cases, such uncertainties distribute at random without concentrating in a certain sample. In other words, variations resulting from such uncertainties and variations in electric characteristics themselves of a sample are independent of each other. Accordingly, assuming that a standard deviation of a measured value is "sgr", a standard deviation resulting from variations in electric characteristics of a sample themselves is "sgr"0, and a standard deviation resulting from uncertainties such as an external noise is "sgr"1, the following relation
"sgr"2="sgr"02+"sgr"12xe2x80x83xe2x80x83(1) 
holds.
As seen from the expression (1), in order that the standard deviation "sgr"0 resulting from variations in electric characteristics of a sample themselves should be reflected on the standard deviation "sgr" of a measured value, it is necessary to minimize a value of the standard deviation "sgr"1 resulting from uncertainties such as an external noise to a negligible extent compared to a value of the standard deviation "sgr"0.
There is a measuring device in which, in order to minimize the value of the standard deviation "sgr"1, plural measurements are automatically made on the same characteristic amount of a sample, and a processing such as weight assignment (e.g., assigning light weights on data measured at the outset of the measurements since an apparatus may have not yet been adjusted completely, and assigning heavier weights on data measured later) is further carried out, thereby obtaining an average value of measured results obtained by the plural measurements and outputting the average value as the measured result of the sample (hereinafter referred to as averaging process). The number of measurements automatically performed in the measuring device (hereinafter referred to as the number of averaging) is a parameter that can be set freely by an observer.
Generally, an influence of uncertainties exerted on a measured result decreases with an increase in the number of averaging, however, the time required for measurements increases with the increase in the number of averaging. Therefore, it is necessary to set a value of the number of averaging performed in the above measuring device to an optimum value in order to maintain compatibleness of required measuring accuracy and efficiency in measurements.
However, there has not been established a method of determining the number of averaging, but the number has been determined empirically.
A first aspect of the present invention is directed to a measuring method of making measurements on a predetermined characteristic amount of a plurality of samples and obtaining a standard deviation of the predetermined characteristic amount of the plurality of samples, wherein averaging process of making measurements by the predetermined number N0 and obtaining an average value of measured results is performed in each one of the measurements. The measuring method comprises the steps of: (a) making plural measurements on the predetermined characteristic amount of one of the plurality of samples under the same condition, thereby calculating a standard deviation "sgr"i of the one of the plurality of samples from measured results of the measurements; (b) making a measurement on the predetermined characteristic amount of each of the plurality of samples under the same condition, thereby calculating a standard deviation "sgr"s of the plurality of samples from the measured results of the measurements; (c) obtaining the number of averaging N using the standard deviations "sgr"i and "sgr"s and the predetermined number N0; and (d) performing again the averaging process by the number of averaging N while measuring the predetermined characteristic amount, and calculating the standard deviation "sgr"s of the plurality of samples from measured results of the measurements made on the plurality of samples.
According to a second aspect of the present invention, in the measuring method of the first aspect, in the step (c), 1/xcex512xc2x7("sgr"i/"sgr"s)2xc2x7N0 is calculated using the ratio of uncertainties xcex51 which is a requested value as a ratio between the standard deviations "sgr"i and "sgr"s, and an integer not lower than a calculated result is employed as the number of averaging N.
According to a third aspect of the present invention, in the measuring method of the second aspect, in the step (c), a value employed as the number of averaging N is obtained by adding a predetermined allowance to the lowest integer not lower than the calculated result of 1/xcex512xc2x7("sgr"i/"sgr"s)2xc2x7N0.
According to a fourth aspect of the present invention, the measuring method of the second aspect further comprises the step of (e) judging before the step (c) whether the standard deviation "sgr"s calculated in the step (b) satisfies a relation "sgr"ixe2x89xa6xcex51xc2x7"sgr"s using the standard deviation "sgr"i and the ratio of uncertainties xcex51 calculated in the step (a), wherein the steps (c) and (d) are performed only when the relation does not hold in the step (e).
A fifth aspect of the present invention is directed to a measuring device comprising: measuring means for measuring a predetermined characteristic amount of a plurality of samples; first calculating means for calculating a standard deviation of the predetermined characteristic amount; second calculating means for calculating the number of averaging N; and controlling means for controlling the measuring means and first and second calculating means, and causing the measuring means to make measurements by the predetermined number N0 in each of the measurements, thereby performing averaging process of obtaining an average value of measured results. In the measuring device, the measuring means makes plural measurements on the predetermined characteristic amount of one of the plurality of samples under the same condition, the controlling means provides the first calculating means with the measured results of the plural measurements made by the measuring means, thereby causing the first calculating means to calculate a standard deviation "sgr"i of the one of the plurality of samples, the measuring means further makes a measurement on the predetermined characteristic amount of each one of the plurality of samples under the same condition, the controlling means provides the first calculating means with each of measured results of the measurements made on the plurality of samples by the measuring means, thereby causing the first calculating means to calculate a standard deviation "sgr"s of the plurality of samples, the controlling means provides the second calculating means with the standard deviations "sgr"i and "sgr"s obtained by the first calculating means and the predetermined number N0, thereby causing the second calculating means to calculate the number of averaging N, the controlling means again performs averaging process on each of the plurality of samples by the number of averaging N, while causing the measuring means to measure the predetermined characteristic amount, and providing the first calculating means with each of measured results of the plurality of samples made by the measuring means, thereby causing the first calculating means to calculate a standard deviation "sgr"s of the plurality of samples based on measurement results of the plurality of samples so as to be outputted.
According to a sixth aspect of the present invention, in the measuring device of the fifth aspect, the second calculating means calculates 1/xcex512xc2x7("sgr"i/"sgr"s)2xc2x7N0 using a ratio of uncertainties xcex51 previously inputted which is a requested value as a ratio between the standard deviations "sgr"i and "sgr"s, and employs an integer not lower than a calculated result as the number of averaging N.
According to a seventh aspect of the present invention, in the measuring device of the sixth aspect, the second calculating means employs, as the number of averaging N, a value obtained by adding a predetermined allowance to the lowest integer not lower than the calculated result of 1/xcex512xc2x7("sgr"i/"sgr"s)2xc2x7N0.
According to an eighth aspect of the present invention, in the measuring device of the sixth aspect, the controlling means judges whether the standard deviation "sgr"s calculated by the first calculating means satisfies a relation "sgr"ixe2x89xa6xcex51xc2x7"sgr"s in which the standard deviation "sgr"i and the ratio of uncertainties xcex51 are used before causing the second calculating means to calculate 1/xcex512xc2x7("sgr"i/"sgr"s)2xc2x7N0, and continues calculation at the first and second calculating means only when the relation does not hold.
With the measuring method of the first aspect, the number of averaging N is obtained using the standard deviations "sgr"i and "sgr"s and the predetermined number of averaging N0 to calculate the standard deviation "sgr"s. Thus, the number of averaging in the averaging process can be determined while maintaining compatibleness of the required measuring accuracy and efficiency in measurements.
With the measuring method of the second aspect, 1/xcex512xc2x7("sgr"i/"sgr"s)2xc2x7N0 is obtained using the ratio of uncertainties xcex51, and the standard deviation "sgr"s is calculated employing an integer not lower than the obtained result as the number of averaging N, which enables to determine the number of averaging N in such a manner that the condition "sgr"i less than  less than "sgr"s holds.
With the measuring method of the third aspect, a value obtained by adding a predetermined allowance Nm to the lowest integer not lower than the obtained result of 1/xcex512xc2x7("sgr"i/"sgr"s)2xc2x7N0 is employed as the number of averaging N, so that an appropriate setting of the predetermined allowance Nm allows to adjust the balance between the required measuring accuracy and efficiency in measurements.
With the measuring method of the fourth aspect, it is judged whether the calculated standard deviation "sgr"s satisfies the relation "sgr"ixe2x89xa6xcex51xc2x7"sgr"s, which enables to determine whether a standard deviation with a sufficient measuring accuracy has been obtained. When the relation holds, later calculations can be omitted.
In the measuring device of the fifth aspect, the number of averaging N is obtained using the standard deviations "sgr"i and "sgr"s and the predetermined number of averaging N0 to calculate the standard deviation "sgr"s. Thus, the number of averaging in the averaging process can be determined while maintaining compatibleness of the required measuring accuracy and efficiency in measurements.
In the measuring device of the sixth aspect, 1/xcex512xc2x7("sgr"i/"sgr"s)2xc2x7N0 is obtained using the ratio of uncertainties xcex51, and the standard deviation "sgr"s is calculated employing an integer not lower than the obtained result as the number of averaging N, which enables to determine the number of averaging N in such a manner that the condition "sgr"i less than  less than "sgr"s holds.
In the measuring device of the seventh aspect, a value obtained by adding a predetermined allowance Nm to the lowest integer not lower than the obtained result of 1/xcex512xc2x7("sgr"i/"sgr"s)2xc2x7N0 is employed as the number of averaging N, so that an appropriate setting of the predetermined allowance Nm allows to adjust the balance between the required measuring accuracy and efficiency in measurements.
In the measuring device of the eighth aspect, it is judged whether the calculated standard deviation "sgr"s satisfies the relation "sgr"ixe2x89xa6xcex51xc2x7"sgr"s, which enables to determine whether a standard deviation with a sufficient measuring accuracy has been obtained. When the relation holds, later calculations can be omitted.
The present invention is directed to provide a measuring device and a measuring method for obtaining a standard deviation of a certain characteristic amount of a plurality of samples, capable of determining the number of averaging in average taking while maintaining compatibleness of required measuring accuracy and efficiency in measurements.
These and other objects, features, aspects and advantages of the present invention will become more apparent from the following detailed description of the present invention when taken in conjunction with the accompanying drawings.